Textbook reference: Geometries and Groups by V. V. Nikulin and I. R. Shafarevich. page 61-63

Problems on geometry corresponding to a uniformly discontinuous group.

5. On the plane, consider two notions (1) and (2) of equivalence of points defined as follows: points A and B are said to be equivalent if (1) A and B are equidistant from some fixed point P; (2) A and B lie on one line in some fixed pencil of parallel lines. Are either of these notions of equivalence given by some group of motions of the plane? What are the resulting sets of equivalent points, and do either of these satisfy condition (8)? Prove that there are geometries corresponding to either notion of equivalence: (1) the geometry of a ray, and (2) that of the line